taylor's theorem

**calculus?!** · August 2nd 2009, 06:47 PM

Use Taylor's Theorem to prove that
$cosx = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!} x^{2k}$

Thanks!

**o_O** · August 2nd 2009, 07:36 PM

The Taylor series for the function $f(x)$ , centred at a = 0, is given by: $f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n + R_n(x)$

where $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}$ is the remainder term ( $c \in (0,x)$ ). To prove that the Taylor series of $f(x)$ converges to $f(x)$ for all $x$ , we must show that the interval of convergence is $\infty$ and $\lim_{n \to \infty} R_n (x) = 0$ .

Coefficients will be for you to find. Use the ratio test to find the interval of convergence.

For the remainder term, notice that: $0 < |R_n (x)| = \left| \frac{\cos^{(n+1)} (c)}{(n+1)!}x^{n+1}\right| {\color{red}\ < \ } \frac{x^{n+1}}{(n+1)!}$

Use squeeze theorem to finish off (recall: $R_n (x) \rightarrow 0$ iff $|R_n (x)| \rightarrow 0)$ )

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